What is the Holy Spirit?
Are the experiences people have with the Holy Spirit proof of the Almighty?
Define proof. Many experiences would definately be proof for all but the most hardcore skeptics. Due to the nature of some of the experiences some cannot be proof to anyone but the person having the experience.
Which experiences are of the Holy Spirit and which are not?
Read the book of Acts and 1 Corinthians 12-14. Tongues is the most common (possibly universal) sign that is given upon being filled or baptized with the Holy Spirit. But the Holy Spirit at times still works on and through people who have not been filled.
Understanding the Scriptures. Some say they learn from the Holy Spirit.
Is it possible that a group of unbelievers without the Holy Spirit could gain understanding people have learned with the Holy Spirit?
The unbeliever's rejection of the Holy Spirit makes this impossible. In any area of understanding, we have to make assumptions before we can reach conclusions. If we reject the founding assumptions (belief in God), we can't reach the conclusions. It is the Holy Spirit alone who gives understanding when one reads the scriptures; otherwise it is "foolishness".
Remembering verses from the Word when you need it.
Definately. This happened to me a few nights ago. I was terribly sad, depressed, and upset over something. I prayed that God would have mercy and help me. When my head hit the pillow, my mind was flooded with scriptures and praises. I don't know when I fell asleep, but I woke up an hour later with a smile on my face almost singing aloud scriptures that I had forgotten. I was extremely surprised at myself, and I didn't know what had happened at first. I knew that just an hour ago I was incredibly depressed, and now through no effort of my own I had recalled all these scriptures and was the happiest person in the world. That had never happened to me before.
Being convicted of sin and repenting.
Being led by the Holy Spirit to help someone (you may or may not know) at the right time without knowing they needed the help.
The ability to have the "fruit of the Spirit is love, joy, peace, longsuffering, gentleness, goodness, faith, Meekness, temperance".
Is it possible for people, without the Spirit, to have all those attributes naturally and/or until they die? Are there any studies?
I would say all except faith as applied towards God.
Then there are the miracles of healing, raising the dead, and speaking and interpreting in other earthly languages.
Some would disgree with slayings, laughing, and babbling tongues being the work of the Holy Spirit
I would. God is a God of order.